## Algebraic & Geometric Topology

### An algorithm to determine the Heegaard genus of simple $3$–manifolds with nonempty boundary

Marc Lackenby

#### Abstract

We provide an algorithm to determine the Heegaard genus of simple $3$–manifolds with nonempty boundary. More generally, we supply an algorithm to determine (up to ambient isotopy) all the Heegaard splittings of any given genus for the manifold. As a consequence, the tunnel number of a hyperbolic link is algorithmically computable. Our techniques rely on Rubinstein’s work on almost normal surfaces, and also on a new structure called a partially flat angled ideal triangulation.

#### Article information

Source
Algebr. Geom. Topol., Volume 8, Number 2 (2008), 911-934.

Dates
Revised: 1 May 2008
Accepted: 2 May 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796849

Digital Object Identifier
doi:10.2140/agt.2008.8.911

Mathematical Reviews number (MathSciNet)
MR2443101

Zentralblatt MATH identifier
1154.57018

Keywords
Heegaard algorithm 3-manifold

#### Citation

Lackenby, Marc. An algorithm to determine the Heegaard genus of simple $3$–manifolds with nonempty boundary. Algebr. Geom. Topol. 8 (2008), no. 2, 911--934. doi:10.2140/agt.2008.8.911. https://projecteuclid.org/euclid.agt/1513796849

#### References

• F Bonahon, Cobordism of automorphisms of surfaces, Ann. Sci. École Norm. Sup. $(4)$ 16 (1983) 237–270
• D B A Epstein, R C Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988) 67–80
• W Jaco, J Rubinstein, Layered-triangulations of $3$–manifolds
• W Jaco, J L Tollefson, Algorithms for the complete decomposition of a closed $3$–manifold, Illinois J. Math. 39 (1995) 358–406
• S Kojima, Polyhedral decomposition of hyperbolic $3$–manifolds with totally geodesic boundary, from: “Aspects of low-dimensional manifolds”, Adv. Stud. Pure Math. 20, Kinokuniya, Tokyo (1992) 93–112
• M Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000) 243–282
• T Li, Heegaard surfaces and measured laminations. II. Non-Haken $3$–manifolds, J. Amer. Math. Soc. 19 (2006) 625–657
• T Li, Heegaard surfaces and measured laminations. I. The Waldhausen conjecture, Invent. Math. 167 (2007) 135–177
• S Matveev, Algorithmic topology and classification of $3$–manifolds, Algorithms and Computation in Math. 9, Springer, Berlin (2003)
• J W Morgan, On Thurston's uniformization theorem for three-dimensional manifolds, from: “The Smith conjecture (New York, 1979)”, Pure Appl. Math. 112, Academic Press, Orlando, FL (1984) 37–125
• C Petronio, J R Weeks, Partially flat ideal triangulations of cusped hyperbolic $3$–manifolds, Osaka J. Math. 37 (2000) 453–466
• J H Rubinstein, Polyhedral minimal surfaces, Heegaard splittings and decision problems for $3$–dimensional manifolds, from: “Geometric topology (Athens, GA, 1993)”, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 1–20
• M Scharlemann, J Schultens, The tunnel number of the sum of $n$ knots is at least $n$, Topology 38 (1999) 265–270
• M Scharlemann, A Thompson, Thin position for $3$–manifolds, from: “Geometric topology (Haifa, 1992)”, Contemp. Math. 164, Amer. Math. Soc. (1994) 231–238
• J Schultens, The classification of Heegaard splittings for (compact orientable surface)$\,\times\, S\sp 1$, Proc. London Math. Soc. $(3)$ 67 (1993) 425–448
• M Stocking, Almost normal surfaces in $3$–manifolds, Trans. Amer. Math. Soc. 352 (2000) 171–207
• J L Tollefson, Involutions of sufficiently large $3$–manifolds, Topology 20 (1981) 323–352